Marine ecologists estimate the reproduction curve for swordfish in a fishing ground to be f(p) = −0.01p^2 + 8p, where p and f(p) are in hundreds. Find the population that gives the maximum sustainable yield, and the size of the yield.
To find the population that gives the maximum sustainable yield and the size of the yield, we need to find the maximum point on the reproduction curve for swordfish.
The reproduction curve is given by the equation f(p) = -0.01p^2 + 8p, where p is the population in hundreds and f(p) is the reproduction rate.
To find the maximum point on the curve, we need to find the value of p at which the first derivative of the curve with respect to p equals zero.
Step 1: Find the first derivative of the reproduction curve f(p).
f'(p) = d/dp (-0.01p^2 + 8p)
= -0.02p + 8
Step 2: Set the first derivative equal to zero and solve for p.
-0.02p + 8 = 0
-0.02p = -8
p = -8 / -0.02
p = 400
Step 3: Check if the value of p found is a maximum or minimum. To do this, we need to find the second derivative of the reproduction curve.
Step 4: Find the second derivative of the reproduction curve f(p).
f''(p) = d^2/dp^2 (-0.01p^2 + 8p)
= -0.02
Since the second derivative f''(p) is a constant and less than zero (-0.02), we can conclude that the value p = 400 is indeed the maximum point on the reproduction curve.
Therefore, the population that gives the maximum sustainable yield is 400 hundreds of swordfish.
To find the size of the yield, we substitute the value of p into the reproduction curve equation.
f(p) = -0.01p^2 + 8p
f(400) = -0.01(400)^2 + 8(400)
= -0.01(160,000) + 3,200
= -1,600 + 3,200
= 1,600
The size of the yield, in this case, is 1,600 swordfish per year.
so, you want maximum f(p)
f'(p) = -9.02p+8
where f'(p)=0 is the maximum population.
of course, you can find the vertex of a parabola without calculus, but that's the sissy way ...